John Spellar said the decision would be a "very welcome Christmas present" for shipyards, engineering and steel industries facing difficulty.
The Warley MP has led calls for the contract to build the three Fleet Solid Support (FSS) ships, believed to be worth more than £1 billion, to go to British firms.
It comes as the Government announced a £16.5 billion boost to defence spending which Mr Spellar, a former defence minister, called "most welcome".
The MP, who is vice-chair of the House of Commons Defence Committee, said in Parliament: "Prime Minister, look if this boost for defence spending is the first fruits of the departure of Dominic Cummings it is most welcome.
"Especially in ensuring we can continue to work effectively alongside our long-term allies and partners the United States – even more so with the welcome arrival of President Biden.
"But can he also ensure that spending is, wherever possible, directed to firms in the UK and that orders are also pulled forward to get British industry moving.
"He can start with the Fleet Solid Support Ships. Tell the Ministry of Defence to send out the invitations to bid – not in some ill-defined spring as they're saying at the moment, but very early in 2021.
"This would be a very welcome Christmas present and new year message – not only for our shipyards, but also for our engineering and steel industries – and their communities."
Prime Minister Boris Johnson, in response, said Mr Spellar "speaks for many [and] he certainly speaks for me in what he says about the Fleet Solid Support Ships".
Mr Johnson said the Defence Secretary was leading a "renaissance" in shipbuilding and will have heard the Warley MP's points "loud and clear".